Probability model selection using information-theoretic optimization criterion

ABSTRACT

A system and method for the discovery and selection of an optimal probability model. Probability model selection can be formulated as an optimization problem with linear order constraints and a non-linear objective function. In one aspect, an algorithm that employs an information-theoretic based approach is used in which a scalable combinatoric search is utilized as defined by an initial solution and null vectors. The theoretical development of the algorithm has led to a property that can be interpreted semantically as the weight of evidence in information theory.

[0001] The present application claims priority from U.S. ProvisionalApplication Serial No. 60/233,569 entitled, “Probability Model SelectionUsing Information-Theoretic Optimization Criterion,” filed on Sep. 19,2000.

BACKGROUND OF THE INVENTION

[0002] 1. Technical Field

[0003] The present invention relates generally to computer processingand more specifically, to a system and method for discovery andselection of an optimal probability model.

[0004] 2. Description of the Related Art

[0005] Probability models with discrete random variables are often usedfor probabilistic inference and decision support. A fundamental issuelies in the choice and the validity of the probability model.

[0006] In statistics, model selection based on information-theoreticcriteria can be dated back to early 1970's when the Akaike InformationCriterion (AIC) was introduced (Akaike, H., (1973), “Information Theoryand an Extension of the Maximum Likelihood Principle,” in Proceedings ofthe 2nd International Symposium of Information Theory, eds. B. N. Petrovand E. Csaki, Budapest: Akademiai Kiado, pp. 267-281). Since then,various information criteria have been introduced for statisticalanalysis. For example, Schwarz information criterion (SIC) (Schwarz, C.,(1978), “Estimating the Dimension of a Model,” The Annals of Statistics,6, pp. 461-464) was introduced to take into account the maximumlikelihood estimate of the model, the number of free parameters in themodel, and the sample size. SIC has been further studied by Chen andGupta (Chen, J. and Gupta, A. K., “Testing and Locating Variance ChangePoints with Application to Stock Prices,” Journal of the AmericanStatistical Association, V.92 (438), Americal Statistical Association,June 1997, pp. 739-747; Gupta, A. K. and Chen, J. (1996), “DetectingChanges of Mean in Multidimensional Normal Sequences with APplicationsto Literature and Geology,”: Computational Statistics, 11:211-221, 1996,Physica-Verlag, Heidelberg) for testing and locating change points inmean and variance of multivariate statistical models with independentrandom variables. Chen further elaborated SIC to change point problemfor regular models. Potential applications on using informationcriterion for model selection to fields such as environmental statisticsand financial statistics are also discussed elsewhere.

[0007] To date, studies in information criteria for model selection havefocused on statistical models with continuous random variables, and inmany cases, with the assumption of iid (independent and identicallydistributed).

[0008] In decision science, the utility of a decision support model maybe evaluated based on the amount of biased information. Let's assume wehave a set of simple financial decision models. Each model manifests anoversimplified relationship among strategy, risk, and return as threeinterrelated discrete binary-valued random variables. The purpose ofthese models is to assist an investor in choosing the type of aninvestment portfolio based on an individual's investment objective;e.g., a decision could be whether one should construct a portfolio inwhich resource allocation is diversified. Let's assume one's investmentobjective is to have a moderate return with relatively low risk. Supposeif a model returns an equal preference on strategies to, or not to,diversify, it may not be too useful to assist an investor in making adecision. On the other hand, a model that is biased towards one strategyover the other may be more informative to assist one in making adecision—even the decision does not have to be the correct one. Forexample, a model may choose to bias towards a strategy based onprobability assessment on strategy conditioned on risk and return.

[0009] In the operations research community, techniques for solvingvarious optimization problems have been discussed extensively. Simplexand Karmarkar algorithms (Borgwardt K. H., 1987, “The Simplex Method, AProbabilistic Analysis,” Springer-Verlag, Berlin; Karmarkar, N., 1984,“A New Polynomial-time Algorithm for Linear Programming,” Combinatorica4 (4), pp. 373-395) are two methods that are constantly being used, andare robust for solving many linear optimization problems. Wright(Wright, S., 1997, “Primal-Dual Interior Point Methods, SIAM, ISBN0-89871-382-X) has written an excellent textbook on primal-dualformulation for the interior point method with different variants ofsearch methods for solving non-linear optimization problems. It wasdiscussed in Wright's book that the primal-dual interior point method isrobust on searching optimal solutions for problems that satisfy KKTconditions with a second order objective function.

[0010] At first glance, it seems that existing optimization techniquescan be readily applied to solve a probability model selection problem.Unfortunately, there are subtle difficulties that make probability modelselection a more challenging optimization problem. First of all, eachmodel parameter in the optimization problem is a joint probability termbounded between 0 and 1. This essentially limits the polytope of thesolution space to be much smaller in comparison to a non-probabilitybased optimization problem with identical set of non-trivial constraints(i.e., those constraints other than 1≧P_(i)≧0).

[0011] In addition, the choice of robust optimization methodologies isrelatively limited for objective functions with a non-linear logproperty; e.g., an objective function based on Shannon InformationCriterion (Shannon, C. E., and Weaver, W., The Mathematical Theory ofCommunication. University of Urbana Press, Urbana, 1972). Primal-dualinterior point is one of the few promising techniques for theprobability model selection problem. However, unfortunately theprimal-dual interior point method requires the existence of an initialsolution, and an iterative process to solve an algebraic system forestimating incrementally revised errors between a current sub-optimalsolution and the estimated global optimal solution. This raises twoproblems. First, the primal-dual formulation requires a naturalaugmentation of the size of the algebraic system to be solved, even ifthe augmented matrix happens to be a sparse matrix. Since the polytopeof the solution space is “shrunk” by the trivial constraints 1≧P_(i)≧0,solving the augmented algebraic system in successive iterations toestimate incremental revised errors is not always possible. Another evenmore fundamental problem is that the convergence of the iterations inprimal-dual interior point method relies on the KKT conditions. Suchconditions may not even exist in many practical model selectionproblems.

[0012] Accordingly, an efficient and accurate technique for selecting anoptimal probability model, while avoiding the limitations and problemsof existing optimization technologies, is highly desirable.

SUMMARY OF THE INVENTION

[0013] The present invention is directed to a system and method forefficient probability model selection. Advantageously, an interiorsearch of a solution space is based on iteratively changing the linearcombination of solution vectors with null vectors (i.e., vectors mappingto zero), via a matrix defining the algebraic problem. Essentially,alternative solutions can be repeatedly derived using already-availableinformation from previously-run algorithms, thus eliminating additionalcomputation costs of entirely new algebra system equations.

[0014] In one aspect of the present invention, a method for selecting aprobability model is provided comprising the steps of: computing aninitial solution for a primal formulation of an optimization problem;identifying null vectors in a matrix of the primal formulation;obtaining multiple solutions by constructing a linear combination of theinitial solution with the null vectors; and determining a local optimalmodel.

[0015] These and other aspects, features, and advantages of the presentinvention will be described or become apparent from the followingdetailed description of the preferred embodiments, which is to be readin connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016]FIG. 1 is an exemplary flow diagram depicting a method fordetermining a local optimal probability model according to an aspect ofthe present invention.

[0017]FIG. 2 is an exemplary flow diagram depicting a method fordetermining an optimal probability model according to an aspect of thepresent invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0018] It is to be understood that the exemplary system modules andmethod steps described herein may be implemented in various forms ofhardware, software, firmware, special purpose processors, or acombination thereof. Preferably, the present invention is implemented insoftware as an application program tangibly embodied on one or moreprogram storage devices. The application program may be executed by anymachine, device or platform comprising suitable architecture. It is tobe further understood that, because some of the constituent systemmodules and method steps depicted in the accompanying Figures arepreferably implemented in software, the actual connections between thesystem components (or the process steps) may differ depending upon themanner in which the present invention is programmed. Given the teachingsherein, one of ordinary skill in the related art will be able tocontemplate or practice these and similar implementations orconfigurations of the present invention.

[0019] The focus of the present invention is on probability models withdiscrete random variables. While the application of the statisticalmodels discussed in the prior art is mainly for statistical inferencebased on statistic hypothesis test, the application of probabilitymodels is for probabilistic inference. The context of probabilisticinference could range, for example, from probability assessment of anevent outcome to identifying the most probable events, or from testingindependence among random variables to identifying event patterns ofsignificant event association.

[0020] In information theory, the amount of biased probabilityinformation can be measured, for example, by means of expected Shannonentropy defined as −Σ_(i)P_(i) Log P_(i). Let (Ω,ℑ,P) be a givenprobability model; where Ω is the sample space, ℑ is a σ-field of setseach of which is a subset of Ω, and P(E) is the probability of an eventE ε ℑ. A linear (in)equality constraint on a probability model can bedefined as, for example, a linear combination of the joint probabilitiesP's in a model. The model selection problem discussed in the presentinvention can be formally formulated, for example, as follows:

[0021] Let M={M_(i): (Ω,ℑ,P) | i=1, 2, . . . } be a set of probabilitymodels where all models share an identical set of primitive eventsdefined as the supremum (the least upper bound) taken over all partitionof Ω. Let C={C_(i): i=1, 2, . . . } be a set of linear (in)equalityconstraints defined over the joint probability of primitive events.Within the space of all probability models bounded by C, the problem ofprobability model selection is to find the model that maximizes expectedShannon entropy.

[0022] It can be shown that the problem of model selection describedabove is actually an optimization problem with linear order constraintsdefined over the joint probability terms of a model, and a non-linearobjective function (defined by −Σ_(i)P_(i) Log P_(i)). It is importantto note an interesting property of the model selection problem justdescribed:

[0023] Property 1: Principle of Minimum Information Criterion—An optimalprobability model is one that minimizes bias, in terms of expectedentropy, in probability assessments that depend on unknown information,while it preserves known biased probability information specified asconstraints in C.

[0024] To overcome the prior art problems, an optimization algorithmtaking a hybrid approach has been developed according to an aspect ofthe present invention. This optimization algorithm follows the spirit ofthe primal-dual interior point method, but deviates from the traditionalapproach in its search technique towards an optimal solution.

[0025] I. Optimization Algorithm for Probability Model Selection

[0026]FIG. 1 illustrates an exemplary flow diagram of a method fordetermining a probability model according to an aspect of the presentinvention. Initially, in step 101, an initial solution is computed for aprimal formulation of an optimization problem. The primal formulation,or algebraic system equations, can be defined, for example, by theprobability constraint set in the form of Ax=b; i.e., each constraint inC accounts for a row in matrix A, and is linear combination of jointprobability terms, with proper slack variable introduced when necessary.

[0027] A typical scenario in probability model selection is experttestimony or valuable information obtained from data analysis expressedin terms of probability constraints. For example, consider the followingcase (Example A) where one is interested in an optimal probability modelwith two binary-valued random variables:$\underset{\_}{{EXAMPLE}\quad A}\quad${X₁ : 0, 1}, {X₂ : 0, 1}, and  P₀ = Pr (X₁ : 0, X₂ : 0), P₁ = Pr (X₁ : 0, X₂ : 1), P₂ = Pr (X₁ : 1, X₂ : 0), P₃ = Pr (X₁ : 1, X₂ : 1),

[0028] Expert testimony:

P(x ₁:0)≦0.65

P ₀ +P ₁ +S=0.65 ∃S≧0

P(x ₂:0)=0.52

P ₀ +P ₂=0.52

Σ _(i) P _(i)=1.0

P ₀ +P ₁ +P ₂ +P ₃=1.0

[0029] Primal formulation: $\begin{matrix}\lceil 11001 \rceil & {\quad {\lceil P_{0} \rceil \quad \lceil 0.65 \rceil}} \\{10100} & {\quad {{P_{1}} = { {0.52}\Leftrightarrow{A\underset{\_}{x}}  = \underset{\_}{b}}}} \\\lfloor 11110 \rfloor & {\quad {{P_{2}}\quad \lfloor 1.00 \rfloor}} \\\quad & {\quad {P_{3}}} \\\quad & {\quad \lfloor S \rfloor}\end{matrix}$

[0030] In general, a probability model with n probability terms, vinequality constraints, and w equality constraints will result in aconstraint matrix A with size (v+w)(n+v). In the example just shown,n=4, v=1, and w=2.

[0031] A feasible solution is preferably obtained for the primalformulation using any available existing solutions, or for example, byany of the numerical methods proposed by Kuenzi, Tzschach, and Zehnder(Kuenzi H. P., Tzschach H. G., and Zehnder C. A., 1971, “NumericalMethods of Mathematical Optimization,” New York, Academic Press).Another feasible solution is preferably obtained by applying, forexample, the Singular Value Decomposition (SVD) algorithm. Then, theinitial solution x is determined by comparing all the available feasiblesolutions and choosing the best one (i.e., the solution which maximizes,for example, expected Shannon entropy).

[0032] The basic idea of the Kuenzi, Tzschach, and Zehnder approach forsolving an algebraic system of linear constraints is to reformulate theconstraint set by introducing (v+w) variables—one for each constraint.Using the above example,

Z ₀=0.65−P ₀ −P ₁ −S

Z ₁=0.52−P ₀ +P ₂

Z ₂=1.0−P ₀ −P ₁ −P ₂ −P ₃

Z ₀≧0 Z ₁≧0 Z ₂≧0

[0033] The above (in)equalities can be thought of as a constraint set ofyet another optimization problem with a cost function Min[Z₀+Z₁+Z₂].Note that a feasible solution of this new optimization problem is avector of seven parameters [Z₀ Z₁ Z₂ P₀ P₁ P₂ P₃]. If the global minimumcan be achieved in this new optimization problem, this is equivalent toZ₀=Z₁=Z₂=0, which in turn gives a feasible solution for the originalproblem. That is, P_(i)s in the global optimal solution [0 0 0P₀ P₁ P₂P₃] is a feasible solution of the original problem.

[0034] In additional to the Kuenzi, Tzschach, and Zehnder approach forsolving the algebraic system of linear constraints, Singular Valuedecomposition (SVD) algorithm can also be applied to obtain anotherfeasible solution. The basic concept of SVD of matrix A is to express Ain the form of UDV^(T). A is a (v+w) by (n+v) matrix. U is a (v+w) by(n+v) orthonormal matrix satisfying U^(T)U=I, where I is an identitymatrix. D is a diagonal matrix with a size of (n+v) by (n+v). Vtranspose (V^(T)) is a (n+v) by (n+v) orthonormal matrix satisfying VV^(T)=I.

[0035] It can be shown a solution to Ax=b is simply, for example,x=VD⁻¹U^(T) b; where D⁻¹D=I. Note that D⁻¹ can be easily constructedfrom D by taking the reciprocal of non-zero diagonal entries of D whilereplicating the diagonal entry from D if an entry in D is zero.

[0036] Next, vectors that correspond to zero entries (i.e,. null vectorsor vectors mapped to zero in a matrix of the primal formulation) areidentified (step 103). For example, column vectors {V_(i) | i=1,2 . . .} from, for example, the by-product of the SVD that correspond to thezero entries in, for example, the diagonal matrix of the SVD of A, areidentified. In other words, the null column vectors have the property ofmapping the null vectors to a null space via the matrix defining theproblem (i.e., Av_(i)=φ; where V_(i) is a null vector). These nullcolumn vectors are referred to as null vectors hereinafter. It is to benoted that the null column vectors can be obtained by methods such asthose for solving eigen-vector/eigen-values, or, in an embodiment of thepresent invention, from the by-product of SVD that corresponds to thezero entries in the diagonal matrix of the SVD of A.

[0037] It can be shown that, for example, whenever there is a zerodiagonal entry d_(i,i)=0 in D of SVD, a linear combination of a solutionvector x with the corresponding i^(th) column vector of V is also asolution to Ax=b. This is because such a column vector of V actually ismapped to a null space through the transformation matrix A; i.e.,AV_(i)=0.

[0038] This enables a search of the optimal probability model along thedirection of the linear combination of the initial solution vector andthe column vectors of V with the corresponding diagonal entry in D equalto zero.

[0039] In step 105, multiple alternative solutions are obtained byconstructing a linear 20 combination of the initial solution with thenull vectors. For example, multiple alternative solutions y are obtainedby constructing the linear combination of the initial solution x withthe V_(i); i.e., Ay=b where y=x+Σ_(i)a_(i)V_(i) for some constantsa_(i).

[0040] An exemplary local optimal solution of the example discussed inExample A above that minimizes Σ_(i)P_(i) log P_(i) (or maximizes−Σ_(i)P_(i) log P_(i)) is shown below:

x=[P ₀ P ₁ P ₂ P ₃]^(T)=[0.2975 0.24 0.2225 0.24]^(T) with Σ_(i) P _(i)log_(e) P _(i)=−1.380067

[0041] It is to be noted that two different approaches (instead of just,for example, SVD) are preferably used to obtain an initial feasiblesolution according to an aspect of the present invention, despite thefact that SVD, for example, generates an initial solution as well as thenecessary information for deriving multiple solutions in a search for anoptimal solution. It is to be noted that the initial feasible solutiondefines the region of the search space the search path traverses.Advantageously, using two different approaches to obtain an initialfeasible solution improves the chance of searching in the space wherethe global optimal solution resides. Further, although SVD is a robustand efficient algorithm for solving linear algebra, the trivialnon-negative constraints (P_(i)≧0) are difficult to include in theformulation required for applying SVD. As a consequence, a solutionobtained from applying, for example, only SVD, albeit satisfying allnon-trivial constraints, may not satisfy the trivial constraints.

[0042] As discussed above, the mechanism for generating multiplesolutions is based on Ay=b where y=x+Σ_(i)a_(i)V_(i) for some constantsa_(i). It is known that SVD may fail to generate a feasible solutionthat satisfies both trivial and non-trivial constraints. When thishappens, however, one can still apply the same mechanism for generatingmultiple solutions using the feasible solution obtained from, forexample, any of the numerical methods of Kuenzi, Tzschach, and Zehnder.

[0043] Next, in step 107 a local optimal model is found through a linearsearch along the vectors that are mapped to the null space (for example,in the SVD process). The local optimal model is the model whichoptimizes a chosen objective function. One of the forms of objectivefunctions used to illustrate the present invention is the Shannoninformation entropy defined by −Σ_(i)P_(i) log P_(i). The local optimalmodel x is a model selected from the multiple alternative solutionsfound in step 105 which optimizes the value of a chosen objectivefunction. In other words, the local optimal model is x=[P₁ . . .P_(n)]^(T), where −Σ_(i)P_(i) log P_(i) of x is the largest among allthe multiple alternative solution models found in step 105. In the case,for example, of maximum Shannon information entropy as an objectivefunction, the local optimal solution is one that minimizes c ^(T) x,where c=[log P₁ . . . log P_(n)]^(T).

[0044] If it is desired to ascertain the optimality of the local optimalmodel (step 109), then the system goes on to step 201 of FIG. 2 below.Otherwise, the system is done (step 113).

[0045]FIG. 2 illustrates an exemplary flow diagram depicting a methodfor determining an optimal probability model according to an aspect ofthe present invention. Initially, in step 201 a dual formulation isconstructed based on the log of the local optimal model determined instep 107. For example, a dual formulation in the form of A^(T) λ≦c canbe constructed.

[0046] Next, an optimal model of the dual formulation is determined(step 203) by solving for λ using, for example, SVD subject tomaximizing an objective function. The objective function is defined bythe linear combination of the constant vector of the primal formulationand the solution of the dual formulation and mathematically has theform, for example, of: b ^(T) λ; where λ=[ log p₁ . . . log p_(n)]^(T).(Note that c is the log of the initial solution (the local optimalmodel) obtained in step 107). This objective function b ^(T) λ can beshown, with a few steps of derivation, as an estimate of an upper boundof the optimality of the local optimal model of step 107.

[0047] Basically, to avoid getting trapped in the local plateau, thepresent invention conducts an optimization in the log space thatcorresponds to the dual part of the model selection problem formulation.Specifically, the constant vector for the algebraic system of the dualpart can be constructed using the local optimal solution obtained instep 107; i.e., $\underset{\_}{{EXAMPLE}\quad B\text{:}}\quad$$\begin{matrix}\lceil 1111000 \rceil & {\quad {\lceil X_{0} \rceil \quad \lceil {\log \quad 0.2975} \rceil}} \\{1010100} & {\quad {{{X_{1}} =  {{\log \quad 0.24}}\quad\Leftrightarrow{{A^{T}\underset{\_}{\lambda}} \leq {\underset{\_}{c}\quad {subject}\quad {to}\quad {maximizing}\quad {\underset{\_}{b}}^{T}\underset{\_}{\lambda}}} };}\quad} \\{0110010} & {\quad {{X_{2}}\quad {{\log \quad 0.2225}}}} \\\lfloor 1010001 \rfloor & {\quad {{S_{0}}\lfloor {\log \quad 0.24} \rfloor}} \\\quad & {\quad {S_{1}}} \\\quad & {\quad {S_{2}}} \\\quad & {\quad \lfloor S_{3} \rfloor}\end{matrix}$${{where}\quad {\underset{\_}{c}}^{T}} = \lbrack {\log \quad 0.2975\quad \log \quad 0.24\quad \log \quad 0.2225\quad \log \quad 0.2} \rbrack$$\quad {{\underset{\_}{b}}^{T} = \lbrack {0.65\quad 0.52\quad 1.0} \rbrack}\quad$

[0048] Note that the column corresponding to the slack variables (i.e.,the last column in A of the primal formulation corresponding to thevariable(s) S) is preferably dropped in the dual formulation since itdoes not contribute useful information to estimating the optimal boundof the solution. In addition, the optimization problem defined in thisdual part comprises linear order constraints and a linear objectivefunction.

[0049] However, certain issues exist in solving the dual part. It is notsufficient just to apply SVD to solve for A^(T) λ≦c because a legitimatesolution requires non-negative values of S_(i)(i=0,1,2,3) in thesolution vector of λ. In the above Example B, although there are fourequations, there are only three variables that can span over the entirerange of real numbers. The remaining four slack variables can only spanover the non-negative range of real numbers. It is not guaranteed therewill always be a solution for the dual part even though there is a localoptimal solution found in the above Example B. For example, the localoptimal solution is listed below:

λ ^(t)=[0.331614 0.10883 7−2.320123] where (maximal) b ^(T) λ=−2.047979

[0050] Next, in step 205, a value of an objective function (for example,b ^(T) λ) due to the global optimal model is compared with the value ofthe objective function (for example, c ^(T) x) due to the local optimalmodel. Then, it is preferably determined if the values are equal (step207). It is to be noted that the optimal value of the objective functionb ^(T) λ is an estimate of the optimality of the solution obtained inthe primal formulation. When c ^(T) x=b ^(T) λ, then the local optimalmodel x is also the global optimal probability model with respect tomaximum expected entropy. Thus, if the values are equal (e.g., whenc^(T)x=b^(T)λ) the best optimal solution has been found and the systemis done (step 211).

[0051] However, it is often the case where c ^(T) x≧b ^(T) λ when thereis a stable solution for the dual part. This can be proved easily withfew steps of derivation similar to that of the standard primal-dualformulation for optimizations described in (Wright 1997). In this case(where, for example, the values are not equal) we can formulate yetanother optimization problem to conduct a search on the optimalsolution. In step 209, an optimal probability model is determined byformulating and solving such another optimization problem. This can bedone, for example, by solving the optimization problem with oneconstraint defined as x ^(T) log x′=b ^(T) λ, with an objective functiondefined as Min|1−Σ_(i)P′_(i)| where Logx′=[ Log P′₁ . . . LogP′_(n)]^(T), where x is the optimal solution vector obtained in the step107. If x′ satisfies all axioms of probability theory, the optimalprobability model to be selected in step 209 is x′. Otherwise, theoptimal probability model to be selected is x.

[0052] If it is desired by the user, for example, to continue searchingfor a better optimal solution (i.e., a model closer to the boundariesdetermined in FIG. 2), the user can go back to step 101 and reformulatethe problem to try to determine a better optimal model. Otherwise, thesystem is done (step 215).

[0053] Note that A is related to the log probability terms, thus thesolution Log x′ represents a log probability model. The concept behind x^(T)_Log x′=b ^(T) λ is to try to get a probability model that has aweighted information measure equal to the estimated global value b ^(T)λ. This involves the following property:

[0054] Property 2: The constraint x ^(T) x′=b ^(T) λ defines asimilarity measure identical to the weight of evidence in comparing twomodels.

[0055] To understand property 2, let's consider the case c ^(T) x≡b ^(T)λ, x ^(T) Log x′=b ^(T) λ becomes x ^(T) Log x′=x ^(T) c, or x^(T)(c−Log x′)=0. x ^(T)(c−Log x′)=0 is equivalent to Σ_(i)P_(i) logP_(i)/P′_(i)=0, which has a semantic interpretation in informationtheory that two models are identical based on the weight of evidencemeasurement function.

[0056] It is to be noted that the optimization in step 209 is a searchin log space, but not necessarily in log probability space since theboundary for the probability space is defined in the objective function,rather than in the constraint set. As a consequence, a solution fromstep 209 does not necessarily correspond to a legitimate candidate forthe probability model selection.

[0057] It is to be noted that the present invention is preferablyimplemented through a time-based software licensing approach asdescribed in co-pending U.S. patent application Ser. No. 60/252,668filed provisionally on Nov. 22, 2000, entitled “Time-Based SoftwareLicensing Approach,” which is commonly assigned and the disclosure ofwhich is hereby incorporated by reference in its entirety.

[0058] In the implementation of an optimization algorithm according tothe present invention, the application also has a feature to supportprobability inference using multiple models. The limitation is that eachquery should be expressed as a linear combination of the jointprobability terms. A probability interval will be estimated for eachquery.

[0059] II. A Practical Example Using a Real World Problem

[0060] Synthetic molecules may be classified as musk-like or notmusk-like. A molecule is classified as musk-like if it has certainchemical binding properties. The chemical binding properties of amolecule depend on its spatial conformation. The spatial conformation ofa molecule can be represented by distance measurements between thecenter of the molecule and its surface along certain rays. This distancemeasurements can be characterized by 165 attributes of continuousvariables.

[0061] A common task in “musk” analysis is to determine whether a givenmolecule has a spatial conformation that falls into the musk-likecategory. Our recent study discovers that it is possible to use only sixdiscretized variables (together with an additional flag) to accomplishthe task satisfactory (with a performance index ranging from 80% to 91%with an average 88%).

[0062] Prior to the model selection process, there is a process ofpattern analysis for selecting the seven variables out of the 165attributes and for discretizing the selected variables. Based on the“musk” data set available elsewhere with 6598 records of 165 attributes,six variables are identified and discretized into binary-valuedvariables according to the mean values. These six variables, referred toas V1 to V6, are from the columns 38, 126, 128, 134, 137, and 165 in thedata file mentioned elsewhere (Murphy 1994). Each of these six randomvariables takes on two possible values {0, 1}. V7 is introduced torepresent a flag. V7:0 indicates an identified pattern is part of aspatial conformation that falls into the musk category, while V7:1indicates otherwise. Below is a list of 14 patterns of variableinstantiation identified during the process of pattern analysis andtheir corresponding probabilities: TABLE 1 Illustration of EventPatterns as Constraints for Probability Model Selection V1 V2 V3 V4 V5V6 V7 Pr(V1, V2, V3, V4, V5, V6, V7) 0 0 0 0 0 0 0 P₀ = 0.03698 0 0 0 00 0 1 P₁ = 0.0565 0 0 0 0 0 1 1 P₃ = 0.0008 0 0 0 0 1 0 0 P₄ = 0.0202 00 0 0 1 0 1 P₅ = 0.0155 0 0 0 0 1 1 1 P₇ = 0.0029 0 0 0 1 0 0 1 P₉ =0.00197 0 0 1 1 1 1 0 P₃₀ = 0.0003 0 1 0 0 0 0 0 P₃₂ = 0.00697 0 1 0 0 00 1 P₃₃ = 0.00318 0 1 0 0 0 1 1 P₃₅ = 0.00136 0 1 0 0 1 0 0 P₃₆ =0.00788 0 1 0 0 1 0 1 P₃₇ = 0.0026 0 1 0 1 0 0 1 P₄₁ = 0.0035

[0063] Remark: The index i of P_(i) in the table shown above correspondsto an integer value whose binary representation is the instantiation ofthe variables (V1 V2 V3 V4 V5 V6 V7).

[0064] A pattern of variable instantiation that is statisticallysignificant may appear as part of the spatial conformation that existsin both the musk and the non-musk categories; for example, the first tworows in the above table. As a result, the spatial conformation of amolecule may be modeled using the probability and statisticalinformation embedded in data to reveal the structure characteristics.One approach to represent the spatial conformation of a molecule is todevelop a probability model that captures the probability informationshown above, as well as the probability information shown below topreserve significant statistical information existed in data:

[0065] P(V1:0)=0.59 P(V2:0)=0.462 P(V3:0)=0.416

[0066] P(V4:0)=0.5215 P(V5:0)=0.42255

[0067] Note that a probability model of musk is defined by a jointprobability distribution of 128 s terms; i.e., P₀ . . . P₁₂₇. In thisexample we have 20 constraints C₀ . . . C₁₉; namely,

[0068] C₀:P₀=0.03698 C₁:P₁=0.0565 C₂:P₃=0.0008 C₃:P₄=0.0202

[0069] C₄:P₅=0.0155 C₅:P₇=0.0029 C₆:P₉=0.00197 C₇:P₃₀=0.0003

[0070] C₈:P₃₂=0.00697 C₉:P₃₃=0.00318 C₁₀:P₃₅=0.00136 C₁₁:P₃₆=0.00788

[0071] C₁₂:P₃₇=0.0026 C₁₃:P₄₁=0.0035

[0072] C₁₄:P(V1:0)=Σ_(V2,V3,V4,V5,V6,V7)P(V1:0,V2,V3,V4,V5,V6,V7)=0.59

[0073] C₁₅:P(V2:0)=Σ_(V1,V3,V4,V5,V6,V7)P(V1,V2:0,V3,V4,V5,V6,V7)=0.462

[0074] C₁₆:P(V3:0)=Σ_(V1,V2,V4,V5,V6,V7)P(V1,V2,V3:0,V4,V5,V6,V7)=0.416

[0075] C₁₇:P(V4:0)=Σ_(V1,V2,V3,V5,V6,V7)P(V1,V2,V3,V4:0,V5,V6,V7)=0.5215

[0076]C₁₈:P(V5:0)=Σ_(V1,V2,V3,V4,V6,V7)P(V1,V2,V3,V4,V5:0,V6,V7)=0.42255

[0077] C₁₉:Σ_(V1,V2,V3,V4,V5,V6,V7)P(V1,V2,V3,V4,V5,V6,V7)=1.0

[0078] The optimal model identified by applying the algorithm discussedin this paper is shown below: TABLE 2 A Local Optimal Probability Modelof Musk P0-P7: 0.03698 0.0565 0.002036 0.0008 0.0202 0.0115 0.0057290.0029  P8-P15: 0.003083 0.00197 0.003083 0.003083 0.006776 0.0067760.006776 0.006776 P16-P23: 0.006269 0.006269 0.006269 0.006269 0.0099630.009963 0.009963 0.009963 P24-P31: 0.007317 0.007317 0.007317 0.0073170.01101 0.01101 0.0003 0.01101 P32-P39: 0.00697 0.00318 0.004879 0.001360.00788 0.0026 0.008572 0.008572 P40-P47: 0.005927 0.0035 0.0059270.005927 0.00962 0.00962 0.00962 0.00962 P48-P55: 0.009113 0.0091130.009113 0.009113 0.012806 0.012806 0.012806 0.012806 P56-P63: 0.010160.01016 0.01016 0.01016 0.013854 0.013854 0.013854 0.013854 P64-P71:0.000497 0.000497 0.000497 0.000497 0.00419 0.00419 0.00419 0.00419P72-P79: 0.001545 0.001545 0.001545 0.001545 0.005238 0.005238 0.0052380.005238 P80-P87: 0.004731 0.004731 0.004731 0.004731 0.008424 0.0084240.008424 0.008424 P88-P95: 0.005779 0.005779 0.005779 0.005779 0.0094720.009472 0.009472 0.009472  P96-P103: 0.003341 0.003341 0.0033410.003341 0.007034 0.007034 0.007034 0.007034 P104-P111: 0.0043880.004388 0.004388 0.004388 0.008081 0.008081 0.008081 0.008081P112-P119: 0.007575 0.007575 0.007575 0.007575 0.011268 0.0112680.011268 0.011268 P120-P127: 0.008622 0.008622 0.008622 0.0086220.012315 0.012315 0.012315 0.012315

[0079] Expected Shannon entropy=−Σ_(i)P_(i) Log₂ P_(i)=6.6792 bits

[0080] III. Evaluation Protocol Design

[0081] In this section the focus will be on a preliminary evaluation ofan ActiveX application. The evaluation was conducted on an Intel Pentium133MHZ laptop with 32M RAM and a hard disk of 420M bytes working space.The laptop was equipped with an Internet Explorer 4.0 web browser withActiveX enabled. In addition, the laptop also had installed S-PLUS 4.5and an add-on commercial tool for numerical optimization NUOPT. Thecommercial optimizer NUOPT was used for comparative evaluation.

[0082] A total of 17 test cases, indexed as C1-C17 listed in Table 3shown in the next section are derived from three sources for acomparative evaluation. The first source is the Hock and Schittkowskiproblem set (Hock, W., and Schittkowski, K., 1980, Lecture Notes inEconomics and Mathematical Systems 187: Test Examples for NonlinearProgramming Codes, Beckmann M. and Kunzi H. P. Eds., Springer-Verlag,Berlin Heidelberg, New York), which is a test set also used by NUOPT forits benchmark testing. The second source is a set of test cases, whichoriginated in real world problems. The third source is a set of randomlygenerated test cases.

[0083] Seven test cases (C1-C7) are derived from the firstsource—abbreviated as STC (Ci) (the ith Problem in the set of StandardTest Cases of the first source). Four test cases originated from realworld problems in different disciplines such as analytical chemistry,medical diagnosis, sociology, and aviation. The remaining six test casesare randomly generated and abbreviated as RTCi (the ith Randomlygenerated Test Case).

[0084] The Hock and Schittkowski problem set is comprised of all kindsof optimization test cases classified by means of four attributes. Thefirst attribute is the type of objective function such as linear,quadratic, or general objective functions. The second attribute is thetype of constraint such as linear equality constraint, upper and lowerbounds constraint etc. The third is the type of the problems whetherthey are regular or irregular. The fourth is the nature of the solution;i.e., whether the exact solution is known (so-called ‘theoretical’problems), or the exact solution is not known (so-called ‘practical’problems).

[0085] In the Hock and Schittkowski problem set, only those test caseswith linear (in)equality constraints are applicable to the comparableevaluation. Unfortunately those test cases need two pre-processingsteps; namely, normalization and normality. These two pre-processingsare necessary because the variables in the original problems are notnecessarily bounded between 0 and 1—an implicit assumption for terms ina probability model selection problem. Furthermore, all terms must beadded to a unity in order to satisfy the normality property, which is anaxiom of the probability theory.

[0086] The second source consists of four test cases. These four cases(C9, C10, C16 and C17) are originated from real world problems. Thefirst case C9 is from census data analysis for studying social patterns.The second case C10 is from analytical chemistry for classifying whethera molecule is a musk-like. The third case C16 is from medical diagnoses.The last one is from aviation, illustrating a simple model ofaerodynamics for single-engine pilot training.

[0087] In addition to the seven “benchmark” test cases and the four testcases from real world problems, six additional test cases (C8, C11-C15)are included for the comparative evaluation. These six cases, indexed byRTCi (the ith randomly generated test case), are generated based on areverse engineering approach that guarantees knowledge of a solution.Note that all seven test cases from the Hock and Schittkowski problemset do not have to have a solution after the introduction of thenormality constraint (i.e., all variables add up to one). Regarding thetest cases originated from the real world problems, there is again noguarantee of the existence of solution(s). As a consequence, theinclusion of these six cases constitutes yet another test source that isimportant for the comparative evaluation.

[0088] IV. Preliminary Comparative Evaluation

[0089] The results of the comparative evaluation are summarized in Table3 below. The first column in the table is the case index of a test case.The second column indicates the source of the test cases. The thirdcolumn is the number of joint probability terms in a model selectionproblem. The fourth column is the number of non-trivial constraints. Ingeneral, the degree of difficulty in solving a model selection problemis proportional to the number of joint probability terms in a model andthe number of constraints.

[0090] The fifth and the sixth columns are the expected Shannon entropyof the optimal model identified by the commercial tool NUOPT and theActiveX application respectively. Recall the objective is to find amodel that is least biased, thus of maximal entropy, with respect tounknown information while preserving the known information stipulated asconstraints. Hence, a model with a greater entropy value is a bettermodel in comparison to one with a smaller entropy value.

[0091] The seventh column reports the upper bound of the entropy of anoptimal model. Two estimated maximum entropies are reported. The firstestimate is derived based on the method discussed earlier (steps 6 and7). The second estimate (in parenthesis) is the theoretical upper boundof the entropy of a model based on Log₂ n; where n is the number ofprobability terms (3^(rd) column) in a model. Further details about thetheoretical upper bound are referred to the report elsewhere (Shannon1972).

[0092] The last column indicates whether an initial guess is providedfor the prototype software to solve a test case. The prototypeimplementation allows a user to provide an initial guess before thealgorithm is applied to solve a test case (e.g., C3b, C7b, C12b, C14b,and C15b). There could be cases where other tools may reach a localoptimal solution that can be further improved. This feature providesflexibility to further improve a local optimal solution. TABLE 3Comparative Evaluation Results Source of test case/ # of non- EntropyWith Application # of trivial NUOPT: entropy Prototype: entropy upperbound initial Case Domain terms constraints of optimal model of optimalmodel estimate guess C1 STC (P55) 6 3 2.5475 2.55465 3.3 No (2.58) C2STC (P21) 6 3 0.971 0.971 *1.306 No (2.58) C3a STC (P76) 4 4 1.98390.9544 7.07 No (2) C3b STC (P76) 1.9855 — Yes C4 STC (P86) 5 8 — — — NoC5 STC (P110) 10 21 — — — No C6 STC (P112) 10 4 3.2457 3.2442 3.966 Nochemical (3.322) equilibrium C7a STC (P119) 16 9 3.498 2.7889 −(4) NoC7b STC (P119) 3.4986 −(4) Yes C8 RTC1 4 3 1.9988 1.991 2.9546 No (2) C9Census Bureau/ 12 10 2.8658 2.8656 −(3.5849) No sociology study C10Chemical 128 20 6.6935 6.6792 23.633 No analysis (Ex. in (7) section 6)C11 RTC2 9 4 2.9936 2.9936 4.247 No (3.167) C12a RTC3 4 3 1.328 0.85545*1.9687 No (2) C12b RTC3 1.328 6.242 Yes (2) C13 RTC4 4 3 2 1.889 3.3589No (2) C14a RTC5 4 3 1.72355 0.971 −(2) No C14b RTC5 1.72355 5.742 Yes(2) C15a RTC6 4 3 1.96289 0.996 −(2) No C15b RTC6 1.96289 6.09755 Yes(2) C16 Medical 256 24 2.8658 3.37018 8.726 No diagnosis (8) C17Single-engine 2187 10 10.13323 10.13357 *11.0406 No pilot training(11.0947) model

[0093] V. Discussion of Comparative Evaluation

[0094] As shown in Table 3, both the prototype implementation of theoptimization algorithm according to the present invention and thecommercial tool NUOPT solved 15 out of the 17 cases. Furtherinvestigation reveals that the remaining two test cases have nosolution. For these 15 cases, both systems are capable of reachingoptimal solutions similar to each other in most of the cases. In onecase (C16) the ActiveX application reached a solution significantlybetter than NUOPT, while NUOPT reached a significantly better solutionin four cases (C3, C12, C14, C15). It is to be noted that the ActiveXapplication actually improves the optimal solution of NUOPT in one ofthese four cases (C3) when the ActiveX application uses the optimalsolution of NUOPT as an initial guess in an attempt to further improvethe solutions of these problems.

[0095] Referring to the seventh column, the result of estimating theupper bound entropy value of the global optimal model using the proposeddual formulation approach is less than satisfactory. In only three(marked with *) of the 15 solvable test cases the proposed dualformulation approach yields a better upper bound in comparison to thetheoretical upper bound that does not consider the constraints of a testcase. Furthermore, in only one of the three cases the estimated upperbound derived by the dual formulation approach is significantly betterthan the theoretical upper bound. This suggests the utility of the dualformulation for estimating an upper bound is limited according to ourtest cases.

[0096] It should also be noted that the proposed dual formulation failsto produce an upper bound in three of the 15 solvable cases (C7, C14,and C15). This is due to the fact that the transpose of the originalconstraint set may turn slack variables in the primal formulation tovariables in the dual formulation that have to be non-negative. However,the SVD cannot guarantee to find solutions that those variables arenon-negative. When the solution derived using SVD contains negativevalues assigned to the slack variables, the dual formulation will failto produce an estimate of the upper bound, which occurred three times inthe 15 solvable test cases.

[0097] In the comparative evaluation, it was chosen not to report thequantitative comparison of the run time performance for two reasons.First, the prototype implementation allows a user to control the numberof iterations indirectly through a parameter that defines the size ofincremental step in the search direction of SVD similar to that of theinterior point method. The current setting, for example, is 100 steps inthe interval of possible bounds in the linear search direction of SVD.When the number of steps is reduced, the speed of reaching a localoptimal solution increases. In other words, one can trade the quality ofthe local optimal solution for the speed in the ActiveX application.Furthermore, if one provides a “good” initial guess, one may be able toafford a large incremental step, which improves the speed, without muchcompromise on the quality of the solution. Therefore, a directcomparative evaluation on the run-time performance would not beappropriate.

[0098] The second reason not to have a direct comparative evaluation ofthe run-time is the need of re-formulating a test case using SIMPLE(System for Interactive Modeling and Programming Language Environment)before NUOPT can “understand” the problem, and hence solve it. SinceNUOPT optimizes its run-time performance by dividing the workload ofsolving a problem into two steps, and only reports the elapsed time ofthe second step, it is not possible to establish an objective ground fora comparative evaluation on the run-time. Nevertheless, the ActiveXapplication solves all the test cases quite efficiently. As typical toany ActiveX deployment, a one-time download of the ActiveX applicationfrom the Internet is typically required.

[0099] VI. Conclusion

[0100] The present invention takes advantage of the by-product, forexample, of the singular value decomposition to efficiently generatemultiple solutions in search of an optimal probability model.Specifically, if a linear set of probability constraints is representedin a matrix form A, A can be decomposed into U^(T)DV using singularvalue decomposition with U and V being normalized orthogonal matrices,and D being a diagonal matrix. If the ith diagonal entry of D is zero,then the ith column vector in V is a by-product of the singular valuedecomposition that is mapped to the null space through A. Such a columnvector is referred to as a null vector. All null vectors, together withan initial solution, can be linearly combined to obtain multiplesolutions through an iterative process to search for the optimal model.

[0101] The present invention advantageously utilizes a scalablecombinatoric linear search as defined by the initial solution and thenull vectors. The search is efficient because of its linearity, and thecombinatorial possible combination of the null vectors with the initialsolution for generating multiple solution models can be adjustedaccording to the computational resources available, as well as theproximity to the true global optimal model. It is to be noted that the“best” values on the choice of the number of the null vectors to be usedare preferably one or two.

[0102] Along the linear search, an initial solution is obtained bysolving the boundary conditions if an inequality probability constraintexists. The present invention includes a parameter (constant C) fordetermining the size of a “jump” step in the search. Specifically, C isa normalized value with, preferably, a best value of 4 for the size ofthe “jump” step, with an optimal normalization scalar that can be easilydetermined by maximal possible change allowed for each element of theinitial feasible solution vector to stay within 0 and 1—a requirementfor satisfying basic probability axioms.

[0103] The present invention, in addition to searching on the boundaryas done by most existing technology such as that based on the Simplexmethod, also searches in the interior of the solution space as done, forexample, by the interior point method. However, the interior search ofthe present invention is based on iteratively changing the value of theslack variables and repeatedly deriving alternative solutions using thealready available information from, for example, singular valuedecomposition without additional computation costs of entirely newalgebra system equations.

[0104] Moreover, it is to be noted that the present invention canutilize an initial solution obtained elsewhere through other methodsbesides those discussed in the present application; thus, the presentinvention can compliment other methods to potentially improve thequality of an optimal solution obtained elsewhere.

[0105] It is to be noted also that the method of the present inventioncan be applied to any model selection problem which provides linearorder constraints and non-linear objective functions, and thus is notrestricted to selecting probability models.

[0106] Although illustrative embodiments of the present invention havebeen described herein with reference to the accompanying drawings, it isto be understood that the present invention is not limited to thoseprecise embodiments, and that various other changes and modificationsmay be affected therein by one skilled in the art without departing fromthe scope or spirit of the present invention. All such changes andmodifications are intended to be included within the scope of theinvention as defined by the appended claims.

What is claimed is:
 1. A method for selecting a probability modelcomprising the steps of: a) computing an initial solution for a primalformulation of an optimization problem; b) identifying null vectors in amatrix of the primal formulation; c) obtaining multiple alternativesolutions by constructing a linear combination of the initial solutionwith the null vectors; and d) determining a local optimal model.
 2. Themethod of claim 1, wherein if it is desired to ascertain an optimalityof the local optimal model, further comprising the steps of:constructing a dual formulation based on the local optimal model;determining an optimal model of the dual formulation; comparing a valueof an objective function of the optimal model of the dual formulationwith a value of an objective function of the local optimal model,wherein if said values are unequal, further comprising the step of:determining an optimal probability model.
 3. The method of claim 1,wherein the local optimal model is a model selected from said multiplealternative solutions which optimizes a chosen objective function. 4.The method of claim 1, wherein the primal formulation comprises theform: Ax=b.
 5. The method of claim 1, wherein the step of computing aninitial solution further comprises the steps of: obtaining a firstsolution using a method proposed by Kuenzi, Tzschach and Zehnder;obtaining a second solution using a Singular Value Decompositionalgorithm; comparing said first and second solutions; and selecting asthe initial solution either of the first or second solutions whichoptimizes a chosen objective function.
 6. The method of claim 1, whereinmultiple alternative solutions y are obtained by constructing saidlinear combination of an initial solution x with null vectors V_(i) suchthat: Ay=b, where y=x+Σ _(i) a _(i) V _(i) for a constant a _(i).
 7. Themethod of claim 2, wherein if a user desires to search for a betteroptimal probability model, further comprising the step of returning tostep (a) of claim
 1. 8. The method of claim 2, wherein the dualformulation comprises the form: A ^(T) λ≦c, where λ=[log p ₁ . . . logp_(n)]^(T).
 9. The method of claim 2, wherein said objective function ofthe global optimal model comprises: b ^(T) λ
 10. The method of claim 2,wherein said objective function of the local optimal model comprises: c^(T) x, where c=[log P₁ . . . log p_(n)]^(T).
 11. A program storagedevice readable by a machine, tangibly embodying a program ofinstructions executable by the machine to perform method steps forselecting a probability model, the method comprising the steps of: a)computing an initial solution for a primal formulation of anoptimization problem; b) identifying null vectors in a matrix of theprimal formulation; c) obtaining multiple alternative solutions byconstructing a linear combination of the initial solution with the nullvectors; and d) determining a local optimal model.
 12. The programstorage device of claim 11, wherein if it is desired to ascertain anoptimality of the local optimal model, further comprising the steps of:constructing a dual formulation based on the local optimal model;determining an optimal model of the dual formulation; comparing a valueof an objective function of the optimal model of the dual formulationwith a value of an objective function of the local optimal model,wherein if said values are unequal, further comprising the step of:determining an optimal probability model.
 13. The program storage deviceof claim 11, wherein the local optimal model is a model selected fromsaid multiple alternative solutions which optimizes a chosen objectivefunction.
 14. The program storage device of claim 11, wherein the primalformulation comprises the form: Ax=b.
 15. The program storage device ofclaim 11, wherein the step of computing an initial solution furthercomprises the steps of: obtaining a first solution using a methodproposed by Kuenzi, Tzschach and Zehnder; obtaining a second solutionusing a Singular Value Decomposition algorithm; comparing said first andsecond solutions; and selecting as the initial solution either of thefirst or second solutions which optimizes a chosen objective function.16. The program storage device of claim 11, wherein multiple alternativesolutions y are obtained by constructing said linear combination of aninitial solution x with null vectors V_(i) such that: Ay=b, where y=x+Σ_(i) a _(i) V _(i) for a constant a _(i).
 17. The program storage deviceof claim 12, wherein if a user desires to search for a better optimalprobability model, further comprising the step of returning to step (a)of claim
 1. 18. The program storage device of claim 12, wherein the dualformulation comprises the form: A ^(T) λ≦c , where λ=[log p ₁ . . . logp_(n)]^(T).
 19. The program storage device of claim 12, wherein saidobjective function of the global optimal model comprises: b ^(T) λ 20.The program storage device of claim 12, wherein said objective functionof the local optimal model comprises: c ^(T) x, where c=[log P₁ . . .log P_(n)]^(T).